The mutilated chessboard problem is a tiling puzzle posed by Max Black in 1946 that asks:
Suppose a standard 8×8 chessboard (or checkerboard) has two diagonally opposite corners removed, leaving 62 squares. Is it possible to place 31 dominoes of size 2×1 so as to cover all of these squares? It is an impossible puzzle: there is no domino tiling meeting these conditions. One proof of its impossibility uses the fact that, with the corners removed, the chessboard has 32 squares of one color and 30 of the other, but each domino must cover equally many squares of each color. More generally, if any two squares are removed from the chessboard, the rest can be tiled by dominoes if and only if the removed squares are of different colors. This problem has been used as a test case for automated reasoning, creativity, and the philosophy of mathematics.
History The mutilated chessboard problem is an instance of domino tiling of grids and polyominoes, also known as "dimer models", a general class of problems whose study in statistical mechanics dates to the work of Ralph H. Fowler and George Stanley Rushbrooke in 1937. Domino tilings also have a long history of practical use in pavement design and the arrangement of tatami flooring. The mutilated chessboard problem itself was proposed by philosopher Max Black in his book Critical Thinking (1946), with a hint at the coloring-based solution to its impossibility. It was popularized in the 1950s through later discussions by Solomon W. Golomb (1954), George Gamow and Marvin Stern (1958), Claude Berge (1958), and Martin Gardner in his Scientific American column "Mathematical Games" (1957). The use of the mutilated chessboard problem in automated reasoning stems from a proposal for its use by John McCarthy in 1964. It has also been studied in cognitive science as a test case for creative insight, Black's original motivation for the problem. In the philosophy of mathematics, it has been examined in studies of the nature of mathematical proof.
Solution The puzzle is impossible to complete. A domino placed on the chessboard will always cover one white square and one black square. Therefore, any collection of dominoes placed on the board will cover equal numbers of squares of each color. But any two opposite squares have the same color: both black or both white. If they are removed, there will be fewer squares of that color and more of the other color, making the numbers of squares of each color unequal and the board impossible to cover. The same idea shows that no domino tiling can exist whenever any two squares of the same color (not just the opposite corners) are removed from the chessboard. Several other proofs of impossibility have also been found. A proof by Shmuel Winograd starts with induction. In a given tiling of the board, if a row has an odd number of squares not covered by vertical dominoes from the previous row, then an odd number of vertical dominoes must extend into the next row. The first row trivially has an odd number of squares (namely, 7) not covered by dominoes of the previous row. Thus, by induction, each of the seven pairs of consecutive rows houses an odd number of vertical dominoes, producing an odd total number. By the same reasoning, the total number of horizontal dominoes must also be odd. As the sum of two odd numbers, the total number of dominoes—vertical and horizontal—must be even. But to cover the mutilated chessboard, 31 dominoes are needed, an odd number. Another method counts the edges of each color around the boundary of the mutilated chessboard. Their numbers must be equal in any tileable region of the chessboard, because each domino has three edges of each color, and each internal edge between dominoes pairs off boundaries of opposite colors. However, the mutilated chessboard has more edges of one color than the other.
If two squares of opposite colors are removed, then the remaining board can always be tiled with dominoes; this result is Gomory's theorem, after mathematician Ralph E. Gomory, whose proof was published in 1973. Gomory's theorem can be proven using a Hamiltonian cycle of the grid graph formed by the chessboard squares. The removal of any two oppositely colored squares splits this cycle into two paths with an even number of squares each. Both of these paths are easy to partition into dominoes by following them. Gomory's theorem is specific to the removal of only one square of each color. Removing larger numbers of squares, with equal numbers of each color, can result in a region that has no domino tiling, but for which coloring-based impossibility proofs do not work.
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